Shadow Removal in an Image Captured by a Vehicle-Based Camera Using an Optimized Oriented Linear Axis

ABSTRACT

A method is provided for removing an illumination generated shadow in a captured image. Each pixel of the captured input image is plotted on a two dimensional logarithmic graph. A linear axis for the plurality of color sets is determined that is substantially orthogonal to a respective illumination direction of each respective color set. A log-chromaticity value of each plotted pixel is projected on the axis. An orientation of the linear axis is selected to minimize an illumination effect and provide optimum separation between each of the respective color sets on the linear axis. Edges in the input image and illumination invariant image domain are identified. The identified edges of the input image are compared to identify edges in the illumination invariant image domain. A determination is made whether a shadow edge is present in response to the comparison. A shadow-reduced image is generated for scene analysis by a vehicle vision-based system.

BACKGROUND OF INVENTION

An embodiment relates generally to vision-based object detection systems.

Illumination conditions such as shadows can cause errors in the vision-based object detection systems. Shadows distort the color of a captured object resulting in ambiguities between (1) edges due to shadows and (2) edges between different entities (e.g., road and landscape). Applications have been used to filter out shadows but prior art systems assume the use of a camera having a high quality imager. Cameras with high quality imagers are expensive with large packaging sizes, and therefore, not practical especially in a mass-produced vehicle based vision system. With the use of high quality imager, the camera sensor is assumed to be narrow-banded and behave like Dirac delta functions in that they have a non-null response only at a single wavelength. However, a low cost imager typically used in vehicle vision based object detection systems does not conform to the narrow-band sensor assumption. Therefore, previous techniques for shadow removal are inapplicable with the use low cost production imagers.

Traditional approaches for shadow removal such as that described a publication in the International Journal of Computer Vision (IJCV) entitled “Entropy Minimization for Shadow Removal”, describe removing a shadow using invariant imaging. The concept is to select an illumination direction that minimizes entropy. Minimizing entropy does attempt to minimize chaotic disorder of the scattering of the different color variations; however, as entropy is minimized, then all or some of the color values within each color set would attempt to project to a single grouping or cluster. As result, the technique described in IJCV only minimizes variations within a color set or multiple color sets. Maximizing inter-variation between color sets is not a feasible result under this approach described, particularly, when numerous color sets are used. In the IJVC publication, only three distinct color sets are utilized. If more than the three color sets shown are utilized and more specifically, color sets have similar color variations, then the distinction between the color sets would be small which would ultimately prevent the identification of shadow edges within the invariant image.

SUMMARY OF INVENTION

An advantage of an embodiment is the reduction of shadows from an image captured by an image capture device that is to be analyzed by a vehicle-based vision sensing system. The shadow-reduction technique optimizes a slope of a linear axis so that intra-variation among color values is minimized and inter-variation between color sets is maximized so that it distinguishes and preserves the color chromaticity of each color set while removing any lighting influence exerted on a respective color set.

An embodiment contemplates a method for removing an illumination generated shadow in a captured image. An input image of a scene is captured by an image capture device. Each pixel of the captured input image is plotted on a two dimensional logarithmic graph. Each pixel is a color variation of one of a plurality of color sets in the logarithmic graph. A linear axis for the plurality of color sets is determined. The linear axis is substantially orthogonal to a respective illumination direction of each respective color set. A log-chromaticity value of each plotted pixel is projected on the axis. Each plotted pixel on the linear axis represents a color value of the respective pixels of the input image mapped to an illumination-invariant image domain. An orientation of the linear axis is selected to minimize an illumination effect in the image and provide an optimum separation between each of the respective color sets projected on the linear axis. Edges in the input image are identified. Edges in the illumination invariant image domain are identified. The identified edges of the input image are compared to identified edges in the illumination invariant image domain. A determination is made whether a shadow edge is present in response to an edge identified in the input image and an absence of a correlating edge in the illumination invariant image domain. A shadow-reduced image is generated for scene analysis by a vehicle vision-based system.

BRIEF DESCRIPTION OF DRAWINGS

FIG. 1 is a plan view of a vehicle-based camera capturing an image of a road.

FIG. 2 is a captured image by an image capture device within a vehicle.

FIG. 3 is a block diagram of the shadow reduction process.

FIG. 4 is a graph of an illumination invariant plot illustrating a linear illumination invariant axis.

FIG. 5 is an exemplary illumination invariant plot utilizing entropy minimization.

FIG. 6 is a graph of an illumination invariant plot utilizing a linear illumination invariant axis optimization technique.

FIG. 7 is a flowchart of a method for reducing a shadow from a captured image.

DETAILED DESCRIPTION

There is shown in FIG. 1, a vehicle 10 traveling along a road 12. A vision-based imaging device 14 captures images of the road forward of the vehicle 10 for detecting images in the feasible region of travel (hereinafter referred to as clear path). The vision-based imaging device 14 is used to detect objects. In a preferred embodiment, the vision-based imaging device 14 is used to identify the clear path or lane markings in the road for systems such as, but not limited to, lane departure warning systems. The vision-based imaging device 14 is preferably mounted in the interior of the vehicle just behind the windshield for capturing events occurring exterior and forward of the vehicle. Although the vision-based imaging device 14 may be used for a variety of functions (e.g., night vision enhancement for the driver), the primary purpose as described herein is for systems that require the recognition of road marking, lane markings, road signs, or other roadway objects. An example of such systems includes, but is not limited to, lane departure warning systems where it is imperative that the system is able to identify where the vehicle is in the roadway for alerting the driver of an unintended lane change.

FIG. 2 illustrates an image captured by the vision-based imaging device on the vehicle. Depending on the brightness and angle of an illumination source, shadows may be cast on objects in the travel path of the vehicle thereby increasing the difficulty of the system to distinguish between an object on the road and a shadow cast on the road. As shown in FIG. 2, a vehicle 16 that is traveling in front of the driven vehicle in addition to light posts 18 and 20 may cast shadows 22, 24, and 26 in the roadway making recognition of lane markers 28 on the road difficult.

An exemplary graphical flow process approach for shadow removal is shown in FIG. 3. In block 30, an image containing a shadow is captured by the image capture device. The shadow is a self-shadow of the person operating the image capture device.

In block 31, an illumination-invariant analysis as will be described later is executed for detecting any shadows in the image. In block 32, the input image is represented in an illumination-invariant image domain. The image represented in the illumination-invariant image domain for graphical purposes is a grey-scale image where color sets are replicated regardless of the illumination conditions or shadows present in the input image. It should be understood that for the purposes of implementing this technique in a vehicle, an actual invariant image is not required to be generated; rather, mathematical analysis, modeling, or other representations may be used to model the image in the illumination-invariant image domain. As is shown, the shadow is removed from the illumination-invariant image domain as a result of the illumination-invariant analysis.

In block 33, the input image and the illumination-invariant image domain are compared for determining where shadows are present in the original input image for constructing a shadow-reduced image.

In block 34, the shadow is removed from the captured input image as a result of the comparison between the gradients of the original input image and the gradients of the illumination-invariant image domain.

FIG. 4 describes a mathematical approach to shadow removal which is described as follows. In object detection and classification for an on-road vehicle, the background and foreground of the captured image are constantly changing. For a single RGB (red, green, blue) image, cast shadows can be removed based on illumination-invariant analysis. If lighting is approximately a Planckian source having Lambertian surfaces imaged by three delta function type sensors, then chromaticity band-ratios (e.g., R/G, B/G for a color 3-band RGB image) may be formed. In addition, a straight line is formed for a log plot of two dimensional {log(R/G), log(B/G)} values for any color set under different lighting conditions. Moreover, every such line for each different color set has a same slope. Therefore, this physics-based concept can be used to derive an illumination-invariant image domain in which color values of a color set map to a same value in the illumination invariant image regardless of lighting conditions (i.e., whether shadows are present or not present).

In addition, object edges formed in the image correspond to changes in a material reflectance. Shadow edges are edges that are in the original image but are absent from an invariant image. A thresholding operation is defined on a gradient representation of the image to identify the shadow edge. Since the threshold shadow edges are noisy, morphological operations are applied to expand the edges and fill in some of the gaps in the shadow edges. Moreover, the identified shadow edges are set to zero for removing the effects of illumination changes. An integration step of each processed channel's gradient image is used to recover shadow-reduced images given up to multiplicative constants which are then estimated in order to obtain the final shadow-reduced color image.

The construction of the shadow-reduced invariant image is discussed herein. A graphical representation using grey-scale imaging is utilized. The technique uses a standard color image as the input whereas the output is an illumination-invariant representation of the image. The illumination-invariant image domain is obtained by projecting its log-chromaticity values on the illumination-invariant direction. To perform this projection, a Lambertian model is used for image formation. An assumption is made that if the surface appearance is equal and independent of the viewing direction (i.e., an ideal diffuse surface), a light source with a spectral power distribution (SPD):E(λ) irradiating on this surface and incident on the camera sensors will lead to a response as follows:

$\begin{matrix} {{R_{k} = {{\int_{w}{{E(\lambda)}{S(\lambda)}{Q_{k}(\lambda)}\ {\lambda}\mspace{14mu} k}} = r}},g,b} & (1) \end{matrix}$

where S(λ) represents the surface reflectance that is defined as the fraction of the incident light that is reflected on the per-wavelength basis, E(λ), the SPD of the illuminant, defines the power emitted by the illuminant as a function of wavelength, Q_(k) (λ) is the spectral sensitivity of the imaging device's k^(th) sensor (where k=r, g, b) specifying the proportion of the light absorbed at each wavelength by the sensors. If the above terms are multiplied and integrated over w, the range of wavelengths to which the sensors have a non-zero response, it gives R_(k) the color value at each pixel in the image.

In order to simplify eq. (1) and derive an invariant representation, aside from the Lambertian surface assumption, two other assumptions are utilized. First, an illumination source is assumed to obey Planck's black-body radiators law. Planck's black-body radiators law states that a perfect spherical radiator when heated at a temperature T emits electromagnetic radiations (e.g., shinnying, glitterings) at specific wavelengths. Examples of Planckian light sources include the sun and the sky which are the illumination sources of most interest in our object detection and classification application. The illumination can then be parameterized by its color temperature T as:

$\begin{matrix} {{E(\lambda)} = {{Ic}_{1}\lambda^{- 5}^{\frac{- c_{2}}{T\; \lambda}}}} & (2) \end{matrix}$

where c₁ and c₂ are constants, and I is the overall intensity of the light.

The second assumption is that the camera sensors are assumed to be narrow-band and to behave like Dirac delta functions in that they have a non-null response only at a single wavelength λ_(k). As a result, the camera sensitivities can be represented by the following equation:

Q _(k)(λ)=q _(k)δ(λ−λ_(k))   (3)

where λ_(k) is the only wavelength at which Q_(k) has a non-null response. With these constraints, the original image formation can be expressed as:

$\begin{matrix} {R_{k} = {{Ic}_{1}\lambda_{k}^{- 5}^{\frac{- c_{2}}{T\; \lambda_{k}}}{S\left( \lambda_{k} \right)}{q_{k}.}}} & (4) \end{matrix}$

For any color in the image, the color may be derived by a combination of RGB color channels (e.g., i=red, blue) The band-ratio 2-vector chromaticity can be generated by dividing two color channels:

$\begin{matrix} {{c_{i} = \frac{R_{i}}{R_{p}}},} & (5) \end{matrix}$

where p is fixed to one color channel (e.g., green), and i indexes over the other two channels (e.g., i=red, blue). The effect of the illumination intensity, I, is removed since it is a constant value at each pixel for all three color channels. Therefore, c_(i) does not depend on the intensity and shading information. The logarithms may then be derived as follows:

$\begin{matrix} {{\rho_{i} = {{\log \left( c_{i} \right)} = {{\log \left( \frac{s_{i}}{s_{g}} \right)} + \frac{e_{i} - e_{g}}{T}}}},{i = r},{b.}} & (6) \\ {\begin{bmatrix} \rho_{r} \\ \rho_{b} \end{bmatrix} = {\begin{bmatrix} {{\log \; s_{r}} - {\log \; s_{g}}} \\ {{\log \; s_{b}} - {\log \; s_{g}}} \end{bmatrix} + {T^{- 1}\begin{bmatrix} {e_{r} - e_{g}} \\ {e_{b} - e_{g}} \end{bmatrix}}}} & (7) \end{matrix}$

where s_(k)=c₁λ_(k) ⁻⁵S(λ_(k))q_(k) and

${e_{k} = {- \frac{c_{2}}{\lambda_{k}}}},$

k=r, g,b. Summarizing the above equation in vector form, the following vector is derived:

$\begin{matrix} {\overset{\_}{\rho} = {\overset{\_}{s} + {\frac{1}{T}{\overset{\_}{e}.}}}} & (8) \end{matrix}$

where s is a 2-vector which depends on the surface being imaged and the camera, but is independent of the illuminant, ē is a 2-vector which is independent of surface being imaged, but depends on the camera. As a result, the illumination color changes (i.e., T varies), and the log-chromaticity vector ρ for a given surface moves along a straight line, which starts from the point s and moves along the direction ē. Moreover, the direction of this straight line depends on properties of the camera, but is independent of the surface and the illuminant.

Eq. (8) also effectively shows that, for the same surface under various Planckian illuminants, the log-chromaticity values fall on a line with slope:

$\begin{matrix} {e = {\begin{bmatrix} {e_{r} - e_{g}} \\ {e_{b} - e_{g}} \end{bmatrix}.}} & (9) \end{matrix}$

Projecting each log-chromaticity value on a direction perpendicular to the illumination slope gives a value in the corresponding image location that depends only on color reflectance which is substantially invariant to the illumination. The generated image is the illumination invariant image.

FIG. 4 shows the invariant image formation process. In FIG. 4, the image has log-chromaticity values for four different surfaces (i.e., color sets). This assumes that perfect narrow-band sensors are utilized under a range of Planckian illuminants. It is clear that the log-chromaticity values for each surface fall along the respective lines 40, 41, 42, and 43 in chromaticity space. These lines have direction e. A direction orthogonal to illumination direction is shown by a solid dark line 44 which is referred to as the illumination-invariant axis. Each log-chromaticity value for a given surface projects to a single point along this line regardless of the illumination under which it is viewed. These points represent the invariant image as defined above.

Once an invariant direction is found relative to the illumination direction of each of the color sets, given a new input image, all the pixels are converted into log-chromaticity space and projected onto the invariant direction.

The objective is to generate a color shadow-reduced image. The original input image which contains shadows was used to derive the shadow-reduced invariant image. The edges which correspond to a shadow can be identified by comparing edges of the original image to those derived from the invariant image. By thresholding the shadow edges and setting gradients of shadows in the original image to zero, gradients that include sharp changes due to illumination effects can be excluded. Lastly, integrating the threshold gradients results in a full-color shadow-reduced image. The following provides a detailed description of the process for obtaining the full-color shadow-reduced image.

The first step is to perform shadow edge mask extraction. The original image contains edges that are induced by surface and illumination transitions, but the invariant image only contains edges relevant to the reference changes of the captured surface that are not caused by the shadow. Therefore, edges of the original image are compared to those derived in the invariant image. A shadow edge is defined to be any edge in the original which is not in the invariant image, which corresponds to illumination changes only. Directional gradients ∇I_(orig) and ∇I_(inv) are calculated separately from the original image and invariant image.

Two thresholds t₁ and t₂ are used to evaluate these two edge maps in order to determine the locations of where the original image has a strong edge, whereas the invariant image has a weak edge. A binary shadow edge is generated as:

$\begin{matrix} {{q_{s}\left( {x,y} \right)} = \begin{Bmatrix} 1 & {{{if}\mspace{14mu} {{\nabla I_{orig}}}} > {t_{1}\mspace{14mu} {and}\mspace{14mu} {{\nabla I_{inv}}}} < t} \\ 0 & {otherwise} \end{Bmatrix}} & (10) \end{matrix}$

Alternatively, shadow edge mask extraction may be performed by comparing the norm difference of the respective gradients. A gradient norm is determined from the input image and a gradient norm is determined from the illumination-invariant image domain. A gradient difference is calculated by subtracting the gradient norm of the illumination-invariant image domain from the gradient norm of the input image. The gradient difference is compared to a threshold for determining whether a shadow edge is present.

An initial shadow edge mask that is first generated is imperfect in that the edge mask contains a number of spurious edges. As a result, a set of morphological operations (e.g., close and dilation operations) are utilized to refine the shadow edges for generating a final shadow edge mask.

The second step is to apply shadow-reduced image integration. Since the shadow edges on gradients correspond to the changing illumination, the shadow edges can be removed in the gradients of the original image by thresholding, which uses the shadow edge mask, as described above, to reduce the illumination effects. As a result, the threshold gradients yield the grayscale representation of one channel which is a shadow-reduced image. The final full-color shadow-reduced image is recovered by combining all the RGB channel grayscale shadow-reduced images. To perform this step, the shadows in the gradient of the original image are removed using the threshold function T_(s).

$\begin{matrix} {{T_{s}\left( {{\nabla I},{q_{s}\left( {x,y} \right)}} \right)} = \begin{Bmatrix} 0 & {{{if}\mspace{14mu} {q_{s}\left( {x,y} \right)}} = 1} \\ {\nabla I} & {otherwise} \end{Bmatrix}} & (11) \end{matrix}$

When a shadow edge is identified, the gradients in the original image are set to zero indicating that there is no change of illumination at this point. After thresholding is applied, the gradients are obtained only where sharp changes are present due to the material changes. The gradient is now integrated in order to recover a shadow-reduced image I′ which does not have a shadow. To accomplish this, Poisson equation is used for problem formulation as follows:

59 ² I′=div(T _(s)(∇I,q _(s)(x, y))).   (12)

On the left side of eq. (13), the Laplacian of the image is represented as follows:

$\begin{matrix} {{\nabla^{2}I^{\prime}} = {\frac{\partial^{2I^{\prime}}}{\partial x^{2}} + {\frac{\partial^{2I^{\prime}}}{\partial y^{2}}.}}} & (13) \end{matrix}$

On the right side of eq. (13), the formula is represented as follows:

${{div}\left( {T_{s}\left( {{\nabla I},{q_{s}\left( {x,y} \right)}} \right)} \right)} = {\frac{\partial{T_{s}\left( {{\nabla I},{q_{s}\left( {x,y} \right)}} \right)}}{\partial x} + \frac{\partial{T_{s}\left( {{\nabla I},{q_{s}\left( {x,y} \right)}} \right)}}{\partial y}}$

Therefore, Poisson equation is solved with homogeneous Neumann boundary conditions so that the gradients at the boundary are set to zero. Solving the above equations for each of the three color channels separately derives an exemplary reconstructed gray-scale image for each channel but with some unknown multiplicative constants. Combining the I′ of all RGB channels together produces a color image where the shadows are removed. In addition, to rectify the unknown multiplicative factors and obtain more realistic image colors, a mapping is applied to each pixel that maps the brightest pixels (e.g. the mean of the top 5% of pixels ordered by intensity) in the recovered image to the corresponding pixels in the original image. In practice, each pixel is assigned a projection value after a pixel is projected onto the illumination-invariant axis.

The process as described above successfully removes shadows when a high-quality imaging device is used. However, considering the cost of incorporating a high-quality imaging device in a vehicle, a more likely scenario is the use of low-cost product camera for object detection and classification. The problem is that the low-cost product camera does not satisfy the narrow-band assumption as the low-cost product camera causes a divergence in the log-chromaticity lines. Moreover, it is impossible to determine the invariant image since no invariant direction can be found. The following paragraphs describe processes for adapting the low-cost product camera to improve the performance of shadow removal for object detection and classification.

In the previous paragraphs, the shadow-removal approach projected the log-chromaticity values of different color sets to a linear space to minimize the variance within each projected color set under various lighting conditions. However, as shown in FIG. 5, color sets having a similar color appearance (e.g., orange and yellow) may overlap when projected on the illumination invariant axis. In FIGS. 5, color sets 50, 51, 52, and 53 are plotted in the logarithmic graph. In FIG. 5, illumination directions 54, 55, 56, and 57 are generated for color sets 50-53, respectively. The resulting illumination-invariant axis is shown at 58. FIG. 5 shows that color set 52 and color set 53 are overlapping when projected onto the illumination-invariant axis such that there is no separation of color variances between color set 52 and color set 53.

FIG. 6 illustrates an enhanced approach to differentiating between color sets having similar color appearances. Each pixel of the captured input image is plotted on a two dimensional logarithmic graph and the color sets are represented by 60-63. It should be understood that each of the pixels plotted in the two dimensional space are located at the same plotted logarithmic coordinates as the pixels shown in FIG. 5. However, the axis and the illumination direction of each color set are re-positioned in contrast to FIG. 5.

In FIG. 6, a linear axis 64 is positioned to achieve greater separation between the projected color sets thereby preserving a substantial difference between the color variance of the color sets. The linear axis 64 not only minimizes the illumination affect of the image, but also maximizes separation of distance between respective color sets projected onto the linear axis 64. Since the illumination directions 60-63 are perpendicular to the linear axis 64, the illumination directions of each color set 60-63 will also be repositioned to maintain a direction that is orthogonal to the linear axis. It should be understood that the re-configured illumination directions 60-63 may not necessarily be a directional line that is a best fit for a respective color set per se (i.e., each repositioned line for a respective color set may not necessarily project through each pixel for each respective color set); however, the re-positioning affords an enhanced color separation between of the pixels when projected onto linear axis 64.

Preferably, the linear axis 64 is positioned until maximum separation is established between each of the color sets. The goal is to achieve minimum intra-variation within the color sets while maximizing inter-variation between the color sets. It should also be understood that the selection of the linear axis 64 does not require that the linear axis 64 be selected and then repositioned to provide the optimum orientation; rather, a machine learning algorithm may be utilized to select at the outset the respective orientation of the linear axis 64 that achieves optimum separation between the color sets projected on the linear axis 64. This may be achieved through machine learning algorithms for multiple-object classification such as, but not limited to, multi-class linear discriminant analysis (LDA). Linear discriminant analysis are methods used in machine learning to determine a linear combination of features which either characterize or separate two or more classes of objects or events by maximizing the square sum of the between-class mean differences and minimizing the sum of the within-class variances. The resulting combination of features may be used as a linear classifier.

FIG. 7 is a method for reducing a shadow from a captured image and achieving enhanced separation of color sets. In step 70, an input image is captured of a scene exterior of a vehicle by an image capture device. The image capture device captures images in color.

In step 71, each pixel <R,G,B> of the captured input image is plotted on a two dimensional logarithmic graph.

In step 72, a linear axis is determined for the color sets of the image, the linear axis is substantially orthogonal to a respective illumination direction of each respective color set.

In step 73, a log-chromaticity value of each plotted pixel is projected onto the linear axis. Each plotted pixel on the axis represents a color value of the respective pixels of the input image mapped to an illumination-invariant image domain. The orientation of the linear axis is selected to provide an optimum separation between the respective color sets projected on the linear axis. That is, the orientation of the linear axis provides a substantial distinction between the color sets projected on the linear axis such that there is no overlap of color variations between the color sets, and each respective projected color set projected on the linear axis is clearly distinguishable from an adjacent color set. Obtaining optimum separation between color sets by positioning the axis may utilize the assistance of a multi-class LDA. This is performed as an offline process. Alternatively, it may be performed on-line in real time.

In step 74, representation of the input image in the illumination-invariant image domain is provided utilizing the projection values of each plotted pixel of the specific color set projected on the linear illumination-invariant kernel. Each color value projected on the non-linear illumination-invariant kernel represents a respective pixel of the invariant image in the illumination-invariant image domain. A gradient of the edges in the illumination invariant image domain is calculated by an operator that includes, but is not limited to a Sobel operator. Moreover, a gradient of the edges in the input image is obtained by the Sobel operator.

In step 75, edges are identified in both the input image and the illumination-invariant image domain.

In step 76, directional gradients of the edges in the input image and the illumination invariant image domain are determined by an operator such as a Sobel operator.

In step 77, the direction gradients of the input image and the direction gradients of the illumination-invariant image domain are compared to at least one threshold. A direction gradient of the input image greater than a first threshold and a direction gradient of the illumination-invariant image less than a second threshold indicates that a shadow edge is present in the input image.

In step 78, if the determination is made that the shadow edge is present, then the shadow-reduced image is generated for scene analysis by a vehicle based vision system. The shadow-reduced image may be generated by reconstructing the input image by replacing the shadow effect in the original input image with the associated color set variation as determined in the invariant image analysis.

If the determination is made that a shadow is not present, then the original input image is provided to the vehicle-based vision system for scene analysis in step 79.

While certain embodiments of the present invention have been described in detail, those familiar with the art to which this invention relates will recognize various alternative designs and embodiments for practicing the invention as defined by the following claims. 

1. A method for removing an illumination generated shadow in a captured image, the method comprising the steps of: (a) capturing an input image of a scene by an image capture device; (b) plotting each pixel of the captured input image on a two dimensional logarithmic graph, each pixel being a color variation of one of a plurality of color sets in the logarithmic graph; (c) determining a linear axis for the plurality of color sets, the linear axis being substantially orthogonal to a respective illumination direction of each respective color set; (e) projecting a log-chromaticity value of each plotted pixel on the axis, wherein each plotted pixel on the linear axis represents a color value of the respective pixels of the input image mapped to an illumination-invariant image domain, wherein an orientation of the linear axis is selected to minimize an illumination effect in the image and provide an optimum separation between each of the respective color sets projected on the linear axis; (f) identifying edges in the input image; (g) identifying edges in the illumination invariant image domain; (h) comparing the identified edges of the input image to identified edges in the illumination invariant image domain; (i) determining whether a shadow edge is present in response to an edge identified in the input image and an absence of a correlating edge in the illumination invariant image domain; and (j) generating a shadow-reduced image for scene analysis by a vehicle vision-based system.
 2. The method of claim 1 wherein selecting an orientation of the linear axis to provide optimum separation between each of the respective color sets includes positioning the linear axis until a maximum separation of distance is obtained between the each of the color sets.
 3. The method of claim 2 wherein selecting an orientation of the linear axis to provide optimum separation includes preserving a separation of color variation within each color set.
 4. The method of claim 1 selecting an orientation of the linear axis to provide optimum separation includes minimizing intra-variations of color sets while maximizing inter-variations between color sets.
 5. The method of claim 1 wherein selecting an orientation or the linear axis to provide optimum separation is achieved utilizing a machine learning technique.
 6. The method of claim 5 wherein the machine learning technique includes a multiple class linear discriminant analysis.
 7. The method of claim 6 wherein the multiple class linear discriminant analysis includes at least two color sets wherein a square sum between the color sets is maximized and the sum within the color sets are minimized.
 8. The method of claim 1 wherein the logarithmic graph includes a logarithmic blue-green axis and a logarithmic red-green axis.
 9. The method of claim 1 wherein identifying edges in the input image and the illumination invariant image domain further comprises the steps of: determining a directional gradient from the input image; determining a directional gradient from the illumination-invariant image domain; and comparing the directional gradient of the input image and the directional gradient of the illumination-invariant image domain to at least one threshold for determining whether a shadow edge is present.
 10. The method of claim 9 wherein comparing the directional gradient of the input image and the directional gradient of the illumination-invariant image domain to at least one threshold further comprises: determining whether the directional gradient of the input image is greater than a first threshold; and determining whether the directional gradient of the illumination-invariant image domain is less a second threshold.
 11. The method of claim 9 wherein comparing the directional gradient of the input image and the directional gradient of the illumination-invariant image domain to at least one threshold further comprises: calculating a gradient difference between the directional gradient of the input image and the directional gradient of the illumination-invariant image domain; and comparing the gradient difference to the at least one threshold.
 12. The method of claim 11 wherein calculating a gradient difference comprises the following steps: determining a gradient norm of the input image; determining a gradient norm of the illumination-invariant image domain; and calculating the gradient difference by subtracting the gradient norm of the illumination-invariant image domain from the gradient norm of the input image. 